Castles in the Air

https://commons.wikimedia.org/wiki/File:Caricature_Vernon_1740.png
Caption: "English Satirical Print - Spain builds castles in the air,
Britain makes commerce her care in the War of Jenkins' Ear."

1. Introduction

It is a term of derision that dates back to the late 16th century. To
accuse someone of building a castle in the air is to accuse them of folly
and fantasy, a wondrous construction that is sustained only by
imagination, but without any proper foundation.

This rebuke derives its power from a commonplace of physics. Nothing can
evade the downward pull of gravity unless somehow there is a counteracting
force that pulls in the opposite direction.

This commonsense of ordinary experience is supported by a standard result
in classical physics. The net effect of gravitational forces on an
ordinary body in a gravitational field is to accelerate its center of mass
downward.

Is it possible to conceive systems that evade these results? If the
system consists of finitely many component parts, interacting by the usual
physical laws, then these results cannot be evaded. If, however, we
consider bodies consisting of infinitely many components, then matters are
otherwise.

The structure to be described here consists of a castle that rests upon a
course of stones. That course rests upon another smaller course of smaller
mass; and so on for an infinity of courses. The structure can be so
contrived that its total mass is finite; it occupies a finite extent of
space; and its matter density remains everywhere finite below some fixed
bound.

There are many forces acting on and in the structure. There are the
external gravitational forces that pull the castle and each course of
stones downward. Then there are the forces that each course of stones
exerts upon the one below it because of the weight each course carries.
Finally there are the reaction forces each course of stones exerts back on
the course of stones above it.

One might expect that on summation all these forces would cancel out and
that we are left with just the net external gravitational forces that must
then accelerate the structure downward.

Because of the infinity of the courses of stones, it is possible to set
up the forces between the stones in such a way that they do not mutually
cancel. Rather there is a residual force. That residual force can be set
to cancel exactly the net gravitational force.

Then the castle and its courses of supporting stones floats in the air.

The literature in supertasks exploits the unexpected behavior of
infinitely many interacting systems to enable a classical system of
masses to spring into life spontaneously. The construction of the castle
in the air depends upon an analogous behavior of infinitely many
components. Rather than producing a dynamical effect where none is
expected, it produces no dynamical effect where one is expected. It is a
static supertask.

2. Distributing the forces so the castle stands

The structure consists of a massive castle in a gravitation field that
imparts a constant downward acceleration to free bodies. The castle rests
on the first course of stones. This course of stones rest on a second
course of stones, and so on for infinitely many courses.

Analogous constructions would allow structures that
are accelerated upward or downwards with any nominated acceleration.

Such a structure may fall freely; or it may remain at rest. If we assume
the case of a structure that remains at rest, we can determine the forces
that must be in place for this to be so and affirm that they conform with
Newtonian physics.

These reaction forces are not generally visible in
Newtonian systems, other than through their effects. They are what
prevents someone standing on a hard floor from falling through the floor.
The routine supposition is that the floor somehow knows just how hard to
push back so that the gravitational force on the person standing is
exactly cancelled. This supposition takes a small liberty since there is
no intelligence in the floor deciding just how hard to press back. Rather,
there is a very slight elasticity in the floor so that a slight
deformation of the floor is resisted by a powerful, opposing elastic
force. The weight of a person produces just such an imperceptible
deformation; and the resulting opposing elastic force holds up the person.

In this static case, each component object of the structure is pulled
downward by a gravitational force and the weight of the objects above.
However the component does not fall. It passes all these forces as a
loading to the next lowest course of stones upon which it rests. This next
course of stones exerts a reaction force back onto it. The reaction force
is equal in magnitude but opposite in direction to the loading forces.

As an aid to visualization, we will imagine the courses of stones to be
connected by very stiff springs. In the configuration described, they are
compressed by just the amount needed to sustain the balanced loading and
reaction forces.

This account is already sufficient to display qualitatively the mechanism
that allows the castle to float in the air. It is:

The castle is supported at
rest by reaction forces from the the first course of
stones.
The first course of stones is supported at rest by
reaction forces from the the second course of stones.
The second course of stones is supported at rest by
reaction forces from the third course of stones.
The third course of stones is supported at rest by
reaction forces from the fourth course of stones.
...

This list continues for each course of stones. Since each course of
stones in the infinite structure is supported at rest, it follows that the
entirety of the structure remains at rest.

This same analysis cannot be applied to the case of a castle with a
finite number of courses of stones. We might try to set up a similar chain
of inferences:

The castle is supported (??)
at rest by reaction forces from the the first course of
stones.
The first course of stones is supported (??) at rest by
reaction forces from the the second course of stones.
The second course of stones is supported (??) at rest by
reaction forces from the third course of stones.
The third course of stones is supported (??) at rest by
reaction forces from the fourth course of stones.
...
The last course of stones is supported (??) at rest by
reaction forces from ... What? There are no further courses of stones
beneath it to provide support.

The failure in the finite structure of the last course of stones to be
supported undoes the analysis. This last course will fall. It follows that
the course above it is unsupported and falls; and so on up the structure
until the castle itself falls.

The essential element in the support of the infinite structure is its
infinity: there is no last course of stones. The mode of failure just seen
for the finite case is eluded.

Nothing in these relations of support requires the courses of stones to
be all the same size. We can require each successive course (including its
springs) to be half the height of the one before it. If we assume the
density of the materials used remains the same, then each successive
course will be half the mass of the one above it.

It follows that the total mass of the structure is finite. If we set the
mass of the first course to unity, that mass is:

mass castle + 1 + 1/2 + 1/4 +
1/8 + ... = mass castle + 2

Correspondingly, setting the height of the first course of stones to
unity, the total height of structure is finite:

height cast + 1 + 1/2 + 1/4 +
1/8 + ... = height castle + 2

The structure can hover in the gravitational field with nothing but empty
space underneath it.

3. Applying Newton's laws

The analysis of the last section is sufficient to display the mechanism
through which the castle is able to stand in the air. Here we see a more
formal analysis. It adds nothing other than a display of formulae that
express the ideas of the last section and a resulting assurance that
Newton's laws are respected.

Once again, the structure consists of a massive castle of mass m0
at vertical position x0 in a gravitation field that
imparts a constant downward acceleration g to free bodies. The
castle rests on the first course of stones of mass m1
at vertical position x1. This course of stones rest
on a second course of mass m2 at vertical position x2,
and so on for infinitely many courses.

The net force fi acting on the ith
object (castle or course of stones) is

(1) f0
= m0a0= m0g + f01
fi =
miai = mig + fi,i-1
+ fi,i+1for i>0

where fik is the force exerted on the ith
object by the kth object. (The only cases we will consider are
those in which i and k are one number separated, so
that k=i-1 or k=i+1.) The
acceleration of each object is ai.

The weight of all the objects above the ith object is
transmitted to it as the loading fi,i-1 which
is the force with which the i-1th object bears down on the ith
object. The ith object exerts a reaction force back on the i-1th
object of fi-1,i

Newton's third law requires that the force with which the ith
object acts on the kth object is equal but opposite in sign to
the force with which the kth object acts on the ith
object. That is:

(2) fik = -fki

All these forces must vanish in case the structure is to remain at rest,
hovering in the air. Then the accelerations must also vanish:

(3) 0 = f0 = f1
= f2 = f3 = ... 0
= a0 = a1 = a2
= a3 = ...

We can find the forces fik that realize this case by
solving equations (1), (2) and (3) iteratively:

If the structure is initialized with these inter-object forces, the
springs will all be compressed by just the amount needed to sustain
these forces. Since the net force on each object is zero, the objects do
not accelerate. They remain separated by the same distances, so that the
springs retain their compression and the forces of (1) remain unchanged
from zero through time.

The structure stands, supported by the forces displayed in (1).

Were there only finitely many courses of stones, N say, then
this distribution (4) of forces would not realize a static system. For
then the forces acting on the component objects would not be given by (1)
but by

The inter-object force distribution (4) does yield an acceleration free
solution (3) when substituted into (5). For both mNg >
0 and fN,N-1 > 0,
so that fN > 0.

4. A center of mass theorem

The analysis of the last section is sufficient to show conformity of the
castle in the air with Newtonian mechanics. However the structure appears
to conflict with a familiar result in mechanics. If we have a body in a
gravitational field, its overall motion is determined by the total
gravitational force acting on it and its total mass. Internal forces
between its components need not be considered. The effect of the
gravitational force is an acceleration of the center of mass, where we
have

total gravitational force = total mass x
acceleration of center of mass

This familiar result is not an independent law, but is a result deduced
from the compounded behavior of the components of the system in a
gravitational field. This section will show that, if we carry out the
corresponding deduction for this infinite structure, the castle can remain
in the air.

The calculation that follows computes the net force on the first N
components. Since these first N components are always a
subsystem of the infinite structure, there are always reaction forces
acting on it from further components and they can be set to cancel out the
gravitational forces acting on these first N components. This
cancellation persists when we take the limit over infinitely many
components, yielding an unaccelerated center of mass for the castle in the
air.

To proceed, we define some intermediate quantities. The mass MN
of the first N components is

This expression can be simplified using (6) and applying (2) to the
sequence of loading and reaction forces so that

(f01 + f10) = (f12
+ f21) = (f23 + f32)
= ... = (fN-1,N + fN,N-1)
= 0

We arrive at

(7) FN = M
(d2/dt2) XN
= MN g + fN,N+1

This result (7) marks the point at which the derivations for a finite
structure and an infinite structure separate.

If the structure is finite and has a total of N components,
then there is no N+1th component and thus no reaction force fN,N+1.
We have:

Center of mass theorem for finite structuresFN = M (d2/dt2)
XN = MNg

One might wonder whether some problem lurks in the
taking of this limit. They are notoriously treacherous. I believe that
this limit is benign. If one thinks otherwise and rejects the definition,
then there is no form of the center of mass theorem applicable to the
infinite case. The threat it posed to the castle in the air is gone.

If the structure is infinite, then there is always an N+1th
component and we can arrive at the theorem for the infinite case by
computing the total net force acting on the infinite structure as

F = Lim
(N → ∞)FN

One possibility is that the castle is in free fall. In that case, all the
forces fik vanish, including fN,N+1,
and the limit gives us:

These last two cases conform with our expectations. The net forces acting
on the finite or infinite structures are just the gravitational forces: MNg
and Mg in the two cases respectively. This net force manifests
as an acceleration of the centers of mass XN and X
respectively.

However, in the infinite static case of a castle in the air, the
inter-object forces are given by (4). From it we read that

That is, there is no net force F acting on the infinite
structure as a whole. The external gravitational forces have been
cancelled exactly by the inter-object forces. The center of mass X
is unaccelerated. Since it is by supposition initially at rest, it remains
so.